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Egyptian Fractions

Stage: 3 Challenge Level:

Why do this problem?

Unit fractions are the first fractions children meet, and here
we discover some very surprising and interesting characteristics of
these familiar numbers. Some of these characteristics were known to
the ancient Egyptians whilst other conjectures are yet to be
proved.

Whilst meeting both old and new mathematical ideas, students
can improve their fluency in addition and subtraction of fractions
and be challenged to generalise and explain their findings.

Possible approach

You could choose to set the scene briefly by asking students what
they know about mathematics throughout history, establishing the
idea that some historical maths is distinctly odd to our modern
view point.

Explain that the ancient Egyptians didn't write fractions with a
numerator greater than 1 but expressed every fraction as the sum of
different unit fractions.

Work through the example of $\frac{2}{3}$ as the problem suggests,
asking students to lengthen each successive row by substituting
each unit fraction by a different pair, using methods the students
met in Keep
It Simple

Establish that we can keep lengthening the expression for any
$\frac{2}{n}$ fraction but what would have been of real value to
the Egyptians would have been a method for expressing these
fractions in the shorteset possible way, i.e. using just two
different unit fractions.

Confirm that this is possible for$\frac{2}{3}$ and then set the
challenge to choose their own $\frac{2}{n}$ fraction and express it
as the sum of just two unit fractions. Any that can't be done can
be written up on the board for the rest of the class to
attempt.

Stop the class, and ask them to step back from number crunching and
share any discoveries. Listen for any generalisations and record
them for discussion.

Students could follow this up by exploring fractions of the
form$\frac{3}{n}$, $\frac{4}{n}$ etc and be challenged to express
them in as short a way as possible.

Key questions

What do we already know that could help?

Possible extension

Some students might wish to undertake research about the Rhind
Mathematical Papyrus.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.